We have seen a relationship between the family of 1D models (REF ) parametrized by \(\phi \)  and the 2D quantum Hall effect (REF ). Such a relationship allows for the prediction of quantized charge transport in temporally-modulated chains, i.e., topological charge pumping [1]}, [2]}, [3]}, [4]}. Here, the time-dependent and adiabatic threading over the parameter \(\phi \)  allows for the 1D family of models to cover the 2D space spanned by both \(\phi \)  and \(\theta \) , cf. Eq. (REF ). This, in similitude to the quantized Hall conductance, leads to a quantized number of charges being pushed across the chain per cycle of \(\phi \)  from 0 to \(2\pi \) , with the quantization originating from the Chern number of the gap. Note that such a topological bulk response is associated with the whole pump cycle, and implies a specific boundary phenomena: at the 0D boundary of the pump, subgap modes cross the topological gap to carry the quantized charge across the system [5]}, [3]}, [7]}, [8]}. This can be thought of as an iteration with \(\phi \)  over the dispersive and chiral 1D edge modes of the 2D quantum Hall effect, see Fig. REF (c).
